1. Lecture 9 Symmetric Matrices Subspaces and Nullspaces
Shang-Hua Teng

2. Matrix Transpose Addition: A+B Multiplication: AB Inverse: A-1
Transpose : A-T

3. Transpose

4. Inner Product and Outer Product

5. Properties of Transpose
End of Page 109: for a transparent proof

6. Ellipses and EllipsoidsR r

7. Later R r
Relating to

8. Symmetric Matrix Symmetric Matrix: A= AT
Graph of who is friend with whom
and its matrix
1;John
2:Alice
4:Anu
3:Feng

9. Symmetric Matrix
B is an m by n matrix

10. Elimination on Symmetric Matrices
If A = AT can be factored into LDU with no row exchange, then U = LT. In other words
The symmetric factorization of a symmetric matrix is A = LDLT

11. So we know Everything about Solving a Linear System
Not quite but Almost
Need to deal with degeneracy (e.g., when A is singular)
Let us examine a bigger issues:
Vector Spaces and Subspaces

12. What Vector Spaces Do We Know So Far
Rn: the space consists of all column (row) vectors with n components

13. Properties of Vector Spaces

14. Other Vector Spaces

15. Vector Spaces Defined by a Matrix
For any m by n matrix A
Column Space:
Null Space:

16. General Linear System
The system Ax =b is solvable if and only if b is in C(A)

17. Subspaces
A subspace of a vector space is a set of vectors (including 0) that satisfies two requirements: if v and w are vectors in the subspace and c is any scalar, then
v+w is in the subspace
cv is in the subspace

18. Subspace of R3 (Z): {(0,0,0)} (L): any line through (0,0,0)
(P): any plane through (0,0,0)
(R3) the whole space
A subspace containing v and w must contain all linear combination cv+dw.

19. Subspace of Rn (Z): {(0,0,…,0)} (L): any line through (0,0,…,0)
(P): any plane through (0,0,…,0) …
(k-subspace): linear combination of any k independent vectors
(Rn) the whole space

20. Subspace of 2 by 2 matrices

21. Express Null Space by Linear Combination
A = [1 1 –2]: x + y -2z = 0
x = -y +2z
Pivot variable
Free variables
Set free variables to typical values
(1,0),(0,1)
Solve for pivot variable: (-1,1,0),(2,0,1)
{a(-1,1,0)+b(2,0,1)}

22. Express Null Space by Linear Combination
Guassian Elimination for finding the linear combination: find an elimination matrix E such that
pivot
EA =
free

23. Permute Rows and Continuing Elimination (permute columns)

24. There must be free variables.
Theorem
If Ax = 0 has more have more unknown than equations (m > n: more columns than rows), then it has nonzero solutions.
There must be free variables.

25. Echelon Matrices
Free variables